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A Generalized Characterization of Commutators of Parabolic Singular Integrals

Published online by Cambridge University Press:  20 November 2018

Steve Hofmann ,
Xinwei Li  and
Dachun Yang
Steve Hofmann
Affiliation:
Department of Mathematics, University of Missouri Columbia Columbia, Missouri 65211, U.S.A., email: [email protected]
Xinwei Li
Affiliation:
Department of Mathematics, Washington University, St. Louis, Missouri 63130-4899 U.S.A., email: [email protected]
Dachun Yang
Affiliation:
Department of Mathematics Washington University St. Louis, Missouri 63130-4899 U.S.A., email: [email protected]
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Abstract

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Let $x\,=\,\left( {{x}_{1}},\,.\,.\,.\,,\,{{x}_{n}} \right)\,\in \,{{\mathbb{R}}^{n}}$ and ${{\delta }_{\text{ }\!\!\lambda\!\!\text{ }}}x\,=\,\left( {{\text{ }\!\!\lambda\!\!\text{ }}^{{{\alpha }_{1}}}}{{x}_{1}},\,.\,.\,.\,,\,{{\text{ }\!\!\lambda\!\!\text{ }}^{{{\alpha }_{n}}}}{{x}_{n}} \right)$, where $\text{ }\lambda \,>\text{0}$ and $1\,\le \,{{\alpha }_{1}}\,\le \,\cdot \,\cdot \,\cdot \,\le \,{{\alpha }_{n}}$. Denote $\left| \alpha \right|\,=\,{{\alpha }_{1}}+\,\cdot \,\cdot \,\cdot \,+{{\alpha }_{n}}$. We characterize those functions $A\left( x \right)$ for which the parabolic Calderón commutator

1

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{T}_{A}}f\left( x \right)\equiv \text{p}\text{.v}\text{.}\int_{{{\mathbb{R}}^{n}}}{K\left( x-y \right)\left[ A\left( x \right)-A\left( y \right) \right]}f\left( y \right)dy$$

is bounded on ${{L}^{2}}\left( {{\mathbb{R}}^{n}} \right)$, where $K\left( {{\delta }_{\text{ }\!\!\lambda\!\!\text{ }}}x \right)\,=\,{{\text{ }\!\!\lambda\!\!\text{ }}^{-\,\left| \alpha \right|\,-\,1}}K\left( x \right)$, $K$ is smooth away fromthe origin and satisfies a certain cancellation property.

Keywords

42B20parabolic singular integralcommutatorparabolic BMO sobolev spacehomogeneous spaceT1-theoremsymbol
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 42 , Issue 4 , 01 December 1999 , pp. 463 - 477
Copyright
Copyright © Canadian Mathematical Society 1999

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